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arXiv:2107.02508

Is there a correlation length in a model with long-range interactions?

Debasis Sadhukhan, Jacek Dziarmaga

Published Jul 6, 2021• 5 citations
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Abstract

Considering an example of the long-range Kitaev model, we are looking for a correlation length in a model with long range interactions whose correlation functions away from a critical point have power-law tails instead of the usual exponential decay. It turns out that quasiparticle spectrum depends on a distance from the critical point in a way that allows to identify the standard correlation length exponent, $ν$. The exponent implicitly defines a correlation length $ξ$ that diverges when the critical point is approached. We show that the correlation length manifests itself also in the correlation function but not in its exponential tail because there is none. Instead $ξ$ is a distance that marks a crossover between two different algebraic decays with different exponents. At distances shorter than $ξ$ the correlator decays with the same power law as at the critical point while at distances longer than $ξ$ it decays faster, with a steeper power law. For this correlator it is possible to formulate the usual scaling hypothesis with $ξ$ playing the role of the scaling distance. The correlation length also leaves its mark on the subleading anomalous fermionic correlator but, interestingly, there is a regime of long range interactions where its short distance critical power-law decay is steeper than its long distance power law tail.

DOI
10.48550/arxiv.2107.02508

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Considering an example of the long-range Kitaev model, we are looking for a correlation length in a model with long range interactions whose correlation functions away from a critical point have power-law tails instead of the usual exponential decay. It turns out that quasiparticle spectrum depends on a distance from the critical point in a way that allows to identify the standard correlation length exponent, ννν. The exponent implicitly defines a correlation length ξξξ that diverges when the critical point is approached. We show that the correlation length manifests itself also in the correlation function but not in its exponential tail because there is none. Instead ξξξ is a distance that marks a crossover between two different algebraic decays with different exponents. At distances shorter than ξξξ the correlator decays with the same power law as at the critical point while at distances longer than ξξξ it decays faster, with a steeper power law. For this correlator it is possible to formulate the usual scaling hypothesis with ξξξ playing the role of the scaling distance. The correlation length also leaves its mark on the subleading anomalous fermionic correlator but, interestingly, there is a regime of long range interactions where its short distance critical power-law decay is steeper than its long distance power law tail.
We propose and analyze a generalization of the Kitaev chain for fermions with long-range p-wave pairing, which decays with distance as a power law with exponent α. Using the integrability of the model, we demonstrate the existence of two types of gapped regimes, where correlation functions decay exponentially at short range and algebraically at long range (α > 1) or purely algebraically (α < 1). Most interestingly, along the critical lines, long-range pairing is found to break conformal symmetry for sufficiently small α. This is accompanied by a violation of the area law for the entanglement entropy in large parts of the phase diagram in the presence of a gap and can be detected via the dynamics of entanglement following a quench. Some of these features may be relevant for current experiments with cold atomic ions.
Long-range interactions exhibit surprising features which have been less explored so far. Here, studying a one-dimensional fermionic chain with long-range hopping and pairing, we discuss some general features associated to the presence of long-range entanglement. In particular, after determining the algebraic decays of the correlation functions, we prove that a long-range quantum mutual information exists if the exponent of the decay is not larger than one. Moreover, we show that the time evolution triggered by a quantum quench between short-range and long-range regions, can be characterized by dynamical quantum phase transitions without crossing any phase boundary. We show, also, that the adiabatic dynamics is dictated by the divergence of a topological length scale at the quantum critical point, clarifying the violation of the Kibble-Zurek mechanism for long-range systems.
The relation between scalar field theories with short-range and long-range exchange or correlations is studied. It is shown to all orders in perturbation theory that the critical exponents of fields and composite operators are continuous functions of the parameters alpha ( xi ) characterising the decay rate of the long-range exchange (correlations) with power-like falloff 1/rd+2-2 alpha (1/rd-2 xi ). The scaling law (d-2 xi )v=2, where v is the correlation length exponent, as well as the Harris criteria (1970) are shown to be exact to all orders in perturbation theory. A discrepancy between two widely used approaches to the crossover problem is pointed out.
We study the role of long-range interactions (more precisely, the long-range superconducting gap term) on the nonequilibrium dynamics considering a long-range ppp-wave superconducting chain in which the superconducting term decays with distance between two sites in a power-law fashion characterized by an exponent $\ensuremath{\alpha}$. We show that the Kibble-Zurek scaling exponent, dictating the power-law decay of the defect density in the final state reached following a slow (in comparison to the time scale associated with the minimum gap in the spectrum of the Hamiltonian) quenching of the chemical potential $\ensuremath{\mu}$ across a quantum critical point, depends nontrivially on the exponent $\ensuremath{\alpha}$ as long as $\ensuremath{\alpha}<2$; on the other hand, for $\ensuremath{\alpha}>2$, we find that the exponent saturates to the corresponding well-known value of 1/21/21/2 expected for the short-range model. Furthermore, studying the dynamical quantum phase transitions manifested in the nonanalyticities in the rate function of the return possibility I(t)I(t)I(t) in subsequent temporal evolution following a sudden change in $\ensuremath{\mu}$, we show the existence of a new region; in this region, we find three instants of cusp singularities in I(t)I(t)I(t) associated with a single sector of Fisher zeros. Notably, the width of this region shrinks as $\ensuremath{\alpha}$ increases and vanishes in the limit $\ensuremath{\alpha}\ensuremath{\rightarrow}2$, indicating that this special region is an artifact of the long-range nature of the Hamiltonian.
We analyze the quantum phases, correlation functions and edge modes for a class of spin-1/2 and fermionic models related to the one-dimensional Ising chain in the presence of a transverse field. These models are the Ising chain with anti-ferromagnetic long-range interactions that decay with distance r as 1 / r α , as well as a related class of fermionic Hamiltonians that generalize the Kitaev chain, where both the hopping and pairing terms are long-range and their relative strength can be varied. For these models, we provide the phase diagram for all exponents α , based on an analysis of the entanglement entropy, the decay of correlation functions, and the edge modes in the case of open chains. We demonstrate that violations of the area law can occur for α ≲ 1 , while connected correlation functions can decay with a hybrid exponential and power-law behavior, with a power that is α -dependent. Interestingly, for the fermionic models we provide an exact analytical derivation for the decay of the correlation functions at every α . Along the critical lines, for all models breaking of conformal symmetry is argued at low enough α . For the fermionic models we show that the edge modes, massless for α ≳ 1 , can acquire a mass for α < 1 . The mass of these modes can be tuned by varying the relative strength of the kinetic and pairing terms in the Hamiltonian. Interestingly, for the Ising chain a similar edge localization appears for the first and second excited states on the paramagnetic side of the phase diagram, where edge modes are not expected. We argue that, at least for the fermionic chains, these massive states correspond to the appearance of new phases, notably approached via quantum phase transitions without mass gap closure. Finally, we discuss the possibility to detect some of these effects in experiments with cold trapped ions.
We study the low-energy properties of the long-range random transverse-field Ising chain with ferromagnetic interactions decaying as a power of the distance. Using variants of the strong-disorder renormalization group method, the critical behavior is found to be controlled by a strong-disorder fixed point with a finite dynamical exponent . Approaching the critical point, the correlation length diverges exponentially. In the critical point, the magnetization shows an -independent logarithmic finite-size scaling and the entanglement entropy satisfies the area law. These observations are argued to hold for other systems with long-range interactions, even in higher dimensions.